preserver_image / affine, shift 0
Image under a real-rootedness preserver. All operations
Parameters (20)
- all
- derivative_polynomial, shift 0 422 proof steps, 410 sequences
- euler_polynomial, shift 0 367 proof steps, 352 sequences
- affine, shift 0 333 proof steps, 318 sequences
- reversal, shift 0 169 proof steps, 164 sequences
- mobius, shift 0 160 proof steps, 160 sequences
- affine, shift -1 5 proof steps, 5 sequences
- hadamard_multiplier, shift 0 5 proof steps, 5 sequences
- affine, shift 1 4 proof steps, 4 sequences
- reversal, shift 1 4 proof steps, 4 sequences
- hadamard_multiplier, shift 2 3 proof steps, 3 sequences
- affine, shift 2 2 proof steps, 2 sequences
- euler_polynomial, shift 1 2 proof steps, 2 sequences
- hadamard_multiplier, shift -1 2 proof steps, 2 sequences
- reversal, shift -1 1 proof steps, 1 sequences
- derivative_of_power_multiple, shift 0 1 proof steps, 1 sequences
- derivative_of_power_multiple, shift 1 1 proof steps, 1 sequences
- derivative_polynomial, shift -1 1 proof steps, 1 sequences
- derivative_polynomial, shift 2 1 proof steps, 1 sequences
- euler_polynomial, shift 2 1 proof steps, 1 sequences
- hadamard_multiplier, shift 1 1 proof steps, 1 sequences
318 sequences
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A126351 → x ↦ x+1
1 · 3, 2 · 10, 13, 4 · 37, 71, 43, 8 … row sums 1, 5, 27, 159, 1025, 7221, 55307, …
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A048176: Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).
A028338 → x ↦ (−1/5)x+11 · -10, 1 · 200, -30, 1 · -6000, 1100, -60, 1 … row sums 1, -9, 171, -4959, 193401, -9476649, …
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A113953 → x ↦ x−10
1 · -10, 1 · 80, -18, 1 · -600, 220, -26, 1 … row sums 1, -9, 63, -405, 2511, -15309, 92583, …
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A120910 → basis A013609
3 · 9, 6 · 27, 36, 12 · 81, 162, 108, 24 … row sums 3, 15, 75, 375, 1875, 9375, 46875, …
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A046716 → basis A007318
1 · 2, 1 · 5, 5, 1 · 16, 25, 11, 1 … row sums 1, 3, 11, 53, 345, 2947, 31411, 400437, …
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A099097 → x ↦ x−3
1 · -9, 3 · 78, -53, 9 · -675, 693, -237, 27 … row sums 1, -6, 34, -192, 1084, -6120, 34552, …
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basis-map coefficients of A113214, shifted → x ↦ x+1
1 · -2, -1 · 4, 4, 1 · -8, -12, -6, -1 … row sums 1, -3, 9, -27, 81, -243, 729, -2187, …
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A099097 → x ↦ x−6
1 · -18, 3 · 318, -107, 9 · -5616, 2844, -480, 27 … row sums 1, -15, 220, -3225, 47275, -693000, …
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A060920: Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).
1 · 2, 1 · 5, 5, 1 · 13, 20, 9, 1 … row sums 1, 3, 11, 43, 171, 683, 2731, 10923, …
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A099097 → x ↦ x−7
1 · -21, 3 · 434, -125, 9 · -8967, 3885, -561, 27 … row sums 1, -18, 318, -5616, 99180, -1751544, …
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A113214 → basis A007318
1 · 3, 1 · 4, 5, 1 · 7, 13, 7, 1 … row sums 1, 4, 10, 28, 76, 208, 568, 1552, …
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A034867: Triangle of odd-numbered terms in rows of Pascal's triangle.
A011973 → x ↦ (1/4)x−1/41 · 2 · 3, 1 · 4, 4 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
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A099040 → x ↦ x−7
1 · -14, 2 · 182, -54, 4 · -2352, 1064, -160, 8 … row sums 1, -12, 132, -1440, 15696, -171072, …
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A011973 → basis A007318 → x ↦ x+1
1 · 1 · 3, 1 · 5, 2 … row sums 1, 1, 4, 7, 19, 40, 97, 217, …
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A178821 → basis A007318
1 · 10, 5 · 60, 60, 15 · 280, 420, 210, 35 … row sums 1, 15, 135, 945, 5670, 30618, 153090, …
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A007318 → (1/2+(1/2)n)θ+1+n → x ↦ x+1
1 · 5, 3 · 18, 21, 6 · 56, 96, 54, 10 … row sums 1, 8, 45, 216, 945, 3888, 15309, 58320, …
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A099093 → x ↦ x−2
1 · -6, 3 · 30, -33, 9 · -144, 252, -144, 27 … row sums 1, -3, 6, -9, 9, 0, -27, 81, …
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A038208 → basis A007318
1 · 4, 2 · 16, 16, 4 · 64, 96, 48, 8 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …